To find the slope and the y-intercept of the equation \( \frac{9 + 27}{2x} = -\frac{3}{2}y \), let's first simplify and rearrange the equation into the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
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Simplify the left side: \[ \frac{9 + 27}{2x} = \frac{36}{2x} = \frac{18}{x} \] Thus, the equation becomes: \[ \frac{18}{x} = -\frac{3}{2}y \]
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Now, clear the fraction by multiplying both sides by \( -2x \): \[ -2 \times 18 = -3y \times x \] \[ -36 = -3xy \] Or equivalently: \[ 3xy = 36 \]
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Solve for \( y \): \[ y = \frac{36}{3x} = \frac{12}{x} \]
Next, we want to write this in a more familiar form; to express it in terms of \( y \): \[ y = \frac{12}{x} \]
This equation doesn’t quite fit the \( y = mx + b \) form because it's a hyperbola rather than a linear equation. Therefore it does not have a traditional slope or y-intercept as defined in linear equations.
Since he's looking for the slope and y-intercept in that context, we can note:
- The equation does not yield a linear relationship. Thus, there is no slope or y-intercept in the conventional sense.
To summarize:
- The slope is undefined.
- There is no y-intercept as it does not represent a linear function.
Final Answer:
- The slope is undefined. The y-intercept does not exist in the linear context.